In this work an energy stable ALE FEM scheme for general free surface flows is derived. Our focus is on liquid droplets in contact with solid surfaces, where wetting phenomenon may be observed. The framework we develop is quite general and may be applied to various setups, such as non–homogeneous or inclined supporting surfaces. By non–homogeneous surface, a surface whose physical and/or chemical properties may vary in space and in time is implied. For such surfaces, droplets in contact exhibit different wetting properties in different configurations. Hence, their behavior may be potentially controlled by surface properties (see, e.g., [1]}, [2]}). Indeed, we investigate a couple of interesting examples in Section , including droplets on non–homogeneous inclined surfaces exhibiting both sliding and rolling kinematics. Mathematical model describing the kinematics of droplets on solid surfaces consists of Navier–Stokes equations governing the fluid flow, coupled with equations for the mesh velocity governing the geometry evolution. Fluid equations have to be subjected to boundary conditions which credibly describe the situation on the interface. This proved to be quite challenging for viscous flows in contact with both solid and gaseous phases. Classical no–slip boundary conditions cannot capture the physics of fluid near the triple contact line (solid–gas–fluid interface). It has been observed that the contact line is able to move despite the apparent no–slip of the fluid at the wall. So called generalized Navier boundary conditions (GNBC) derived from the molecular dynamics (MD) theory for the Navier–Stokes equations have been proven to capture the dynamics of the triple contact line adequately (see [3]}, [4]}, [5]}, [6]}). GNBC have already been successfully applied for simulating free surface flows with ALE FEM for various multiphysics phenomena such as free surface flows in [7]}, sliding/rolling droplets kinematics in [8]}, [9]}, droplets kinematics on non–isothermal surfaces in [10]}, chemotaxis phenomena in [11]}, [12]}, to name a few. The main goal of this paper is the derivation of the energy stable numerical scheme based on the mathematical model incorporating GNBC and not the investigation of GNBC themselves. However, we do perform some numerical validation of GNBC within the ALE FEM since they weren't originally developed within this framework.
